VoC Analyses Part 3: Analyze Reinforcement-Learning Results

AIC analyses

Average AIC by age group

Average AIC

AIC difference from best model

Age-related change in parameter estimates from models

Run regressions between model parameters and age

## 
## Call:
## lm(formula = LL ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -146.117  -38.809    1.221   37.467  140.946 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -252.460     26.180  -9.643 1.58e-15 ***
## age            3.244      1.407   2.305   0.0234 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 63.92 on 90 degrees of freedom
## Multiple R-squared:  0.05576,    Adjusted R-squared:  0.04527 
## F-statistic: 5.315 on 1 and 90 DF,  p-value: 0.02344

## 
## Call:
## lm(formula = alphaPosChoice ~ age, data = model_params)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.2998 -0.1843 -0.1090  0.1042  0.8076 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.075584   0.115998   0.652    0.516
## age         0.009314   0.006235   1.494    0.139
## 
## Residual standard error: 0.2832 on 90 degrees of freedom
## Multiple R-squared:  0.0242, Adjusted R-squared:  0.01335 
## F-statistic: 2.232 on 1 and 90 DF,  p-value: 0.1387

## 
## Call:
## lm(formula = alphaNegChoice ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.12509 -0.11233 -0.09502 -0.03547  0.85745 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.150066   0.095837   1.566    0.121
## age         -0.002127   0.005151  -0.413    0.681
## 
## Residual standard error: 0.234 on 90 degrees of freedom
## Multiple R-squared:  0.001892,   Adjusted R-squared:  -0.009199 
## F-statistic: 0.1706 on 1 and 90 DF,  p-value: 0.6806

## 
## Call:
## lm(formula = alphaPosComp ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.15755 -0.13896 -0.11970 -0.00461  0.84894 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.113350   0.100912   1.123    0.264
## age         0.001771   0.005424   0.327    0.745
## 
## Residual standard error: 0.2464 on 90 degrees of freedom
## Multiple R-squared:  0.001183,   Adjusted R-squared:  -0.009914 
## F-statistic: 0.1066 on 1 and 90 DF,  p-value: 0.7448

## 
## Call:
## lm(formula = alphaNegComp ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.18332 -0.18145 -0.15366  0.05692  0.80513 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.816e-01  1.175e-01   1.545    0.126
## age         8.019e-05  6.318e-03   0.013    0.990
## 
## Residual standard error: 0.287 on 90 degrees of freedom
## Multiple R-squared:  1.79e-06,   Adjusted R-squared:  -0.01111 
## F-statistic: 0.0001611 on 1 and 90 DF,  p-value: 0.9899

## 
## Call:
## lm(formula = betaAgency ~ age, data = model_params)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.9275 -3.8227 -0.5232  2.4939 18.7135 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   5.0872     2.2647   2.246   0.0271 *
## age           0.2358     0.1217   1.937   0.0558 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.53 on 90 degrees of freedom
## Multiple R-squared:  0.04003,    Adjusted R-squared:  0.02937 
## F-statistic: 3.753 on 1 and 90 DF,  p-value: 0.05584

## 
## Call:
## lm(formula = betaMachine ~ age, data = model_params)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.3482 -3.1200 -0.6171  2.0051 16.4548 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  5.79744    2.05294   2.824  0.00584 **
## age          0.09143    0.11035   0.829  0.40955   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.013 on 90 degrees of freedom
## Multiple R-squared:  0.00757,    Adjusted R-squared:  -0.003457 
## F-statistic: 0.6865 on 1 and 90 DF,  p-value: 0.4096

## 
## Call:
## lm(formula = agencyBonus ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.82471 -0.15345 -0.04041  0.04863  1.74151 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.132596   0.171398   0.774    0.441
## age         0.010238   0.009213   1.111    0.269
## 
## Residual standard error: 0.4185 on 90 degrees of freedom
## Multiple R-squared:  0.01354,    Adjusted R-squared:  0.002575 
## F-statistic: 1.235 on 1 and 90 DF,  p-value: 0.2694

Plot relations between model parameters and age

Parameter summary statistics

Mixed-effects beta analysis

## Mixed Model Anova Table (Type 3 tests, S-method)
## 
## Model: estimate ~ ageZ * betaType + (1 | subID)
## Data: betas
##          Effect       df        F p.value
## 1          ageZ 1, 90.00     2.73    .102
## 2      betaType 1, 90.00 10.76 **    .001
## 3 ageZ:betaType 1, 90.00     1.41    .238
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: estimate ~ ageZ * betaType + (1 | subID)
##    Data: data
## 
## REML criterion at convergence: 1109.9
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -2.3424 -0.4713 -0.1507  0.4003  3.2096 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  subID    (Intercept) 12.61    3.551   
##  Residual             15.24    3.904   
## Number of obs: 184, groups:  subID, 92
## 
## Fixed effects:
##                Estimate Std. Error      df t value Pr(>|t|)    
## (Intercept)      8.3862     0.4689 90.0000  17.883  < 2e-16 ***
## ageZ             0.7771     0.4702 90.0000   1.653  0.10191    
## betaType1        0.9439     0.2878 90.0000   3.280  0.00148 ** 
## ageZ:betaType1   0.3429     0.2886 90.0000   1.188  0.23790    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) ageZ  btTyp1
## ageZ        0.000              
## betaType1   0.000  0.000       
## ageZ:btTyp1 0.000  0.000 0.000

Predictor	Estimates	SE	Statistic	p
intercept	8.39	0.47	17.88	<0.001
age	0.78	0.47	1.65	0.100
decision stage	0.94	0.29	3.28	0.001
age x decision stage	0.34	0.29	1.19	0.236

Beta plot

Mixed-effects learning rate analysis

## Mixed Model Anova Table (Type 3 tests, S-method)
## 
## Model: estimate ~ ageZ * valence * agency + (1 | subID)
## Data: learning_rates
##                Effect        df        F p.value
## 1                ageZ  1, 90.00     0.52    .473
## 2             valence 1, 270.00   3.07 +    .081
## 3              agency 1, 270.00     0.25    .618
## 4        ageZ:valence 1, 270.00     1.36    .245
## 5         ageZ:agency 1, 270.00     0.22    .637
## 6      valence:agency 1, 270.00 10.03 **    .002
## 7 ageZ:valence:agency 1, 270.00     0.75    .388
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: estimate ~ ageZ * valence * agency + (1 | subID)
##    Data: data
## 
## REML criterion at convergence: 107.5
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -1.3019 -0.5803 -0.3735  0.0936  3.2313 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  subID    (Intercept) 0.003877 0.06226 
##  Residual             0.065636 0.25619 
## Number of obs: 368, groups:  subID, 92
## 
## Fixed effects:
##                         Estimate Std. Error         df t value Pr(>|t|)    
## (Intercept)             0.170801   0.014849  90.000000  11.502  < 2e-16 ***
## ageZ                    0.010716   0.014869  90.000000   0.721  0.47299    
## valence1               -0.023387   0.013355 269.999999  -1.751  0.08105 .  
## agency1                 0.006673   0.013355 269.999999   0.500  0.61771    
## ageZ:valence1          -0.015570   0.013373 269.999999  -1.164  0.24534    
## ageZ:agency1            0.006325   0.013373 269.999999   0.473  0.63660    
## valence1:agency1       -0.042297   0.013355 269.999999  -3.167  0.00172 ** 
## ageZ:valence1:agency1  -0.011560   0.013373 269.999999  -0.864  0.38812    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) ageZ  valnc1 agncy1 agZ:v1 agZ:g1 vln1:1
## ageZ        0.000                                          
## valence1    0.000  0.000                                   
## agency1     0.000  0.000 0.000                             
## ageZ:valnc1 0.000  0.000 0.000  0.000                      
## ageZ:agncy1 0.000  0.000 0.000  0.000  0.000               
## vlnc1:gncy1 0.000  0.000 0.000  0.000  0.000  0.000        
## agZ:vlnc1:1 0.000  0.000 0.000  0.000  0.000  0.000  0.000

## 
##  Paired t-test
## 
## data:  model_params$alphaPosChoice and model_params$alphaNegChoice
## t = 3.2464, df = 91, p-value = 0.001636
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  0.05098873 0.21174803
## sample estimates:
## mean difference 
##       0.1313684

## 
##  Paired t-test
## 
## data:  model_params$alphaPosComp and model_params$alphaNegComp
## t = -0.8713, df = 91, p-value = 0.3859
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  -0.12404217  0.04840164
## sample estimates:
## mean difference 
##     -0.03782026

Predictor	Estimates	SE	Statistic	p
intercept	0.17	0.01	11.50	<0.001
age	0.01	0.01	0.72	0.472
valence	-0.02	0.01	-1.75	0.081
agency	0.01	0.01	0.50	0.618
age x valence	-0.02	0.01	-1.16	0.245
age x agency	0.01	0.01	0.47	0.637
valence x agency	-0.04	0.01	-3.17	0.002
age x valence x agency	-0.01	0.01	-0.86	0.388

Learning rate plot

Relation between parameter estimates and ‘model-free’ regressions

## 
## Call:
## lm(formula = `(Intercept)` ~ agencyBonus, data = voc_REs_RL)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.8973 -0.5048 -0.0711  0.4379  3.2340 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.7368     0.1199  -6.142 2.17e-08 ***
## agencyBonus   2.2749     0.2291   9.928 4.03e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9159 on 90 degrees of freedom
## Multiple R-squared:  0.5227, Adjusted R-squared:  0.5174 
## F-statistic: 98.57 on 1 and 90 DF,  p-value: 4.026e-16

## 
## Call:
## lm(formula = zVoC ~ betaAgency, data = voc_REs_RL)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.86135 -0.30842 -0.04316  0.23221  1.17101 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.833300   0.090307  -9.227 1.16e-14 ***
## betaAgency   0.086421   0.008306  10.405  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4447 on 90 degrees of freedom
## Multiple R-squared:  0.546,  Adjusted R-squared:  0.541 
## F-statistic: 108.3 on 1 and 90 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = zVoC ~ betaAgency + age, data = voc_REs_RL)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.85984 -0.32425 -0.02079  0.25295  1.14000 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.979619   0.187387  -5.228 1.12e-06 ***
## betaAgency   0.084907   0.008487  10.004 3.14e-16 ***
## age          0.008918   0.010004   0.891    0.375    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4452 on 89 degrees of freedom
## Multiple R-squared:  0.5501, Adjusted R-squared:  0.5399 
## F-statistic:  54.4 on 2 and 89 DF,  p-value: 3.676e-16

## 
## Call:
## lm(formula = zVoC ~ betaAgency + betaMachine, data = voc_REs_RL)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.92713 -0.32224 -0.06426  0.27199  1.17530 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.878039   0.097696  -8.987 3.99e-14 ***
## betaAgency   0.081312   0.009342   8.704 1.54e-13 ***
## betaMachine  0.012416   0.010478   1.185    0.239    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4437 on 89 degrees of freedom
## Multiple R-squared:  0.5531, Adjusted R-squared:  0.543 
## F-statistic: 55.07 on 2 and 89 DF,  p-value: 2.721e-16

Questionnaire relations

DOC

## 
## Call:
## lm(formula = DOC ~ zAge, data = DOC)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -32.234  -6.388  -0.270   7.449  30.317 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   95.527      1.255   76.11   <2e-16 ***
## zAge           2.446      1.274    1.92    0.058 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.97 on 89 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.03978,    Adjusted R-squared:  0.02899 
## F-statistic: 3.687 on 1 and 89 DF,  p-value: 0.05804

## 
## Call:
## lm(formula = DOC ~ zBetaAgency * zAgencyBonus * zAge, data = DOC)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -32.177  -6.694   0.498   6.836  28.152 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                    96.3552     1.4448  66.689   <2e-16 ***
## zBetaAgency                    -0.6198     1.5563  -0.398    0.691    
## zAgencyBonus                    0.7234     3.8298   0.189    0.851    
## zAge                            3.1628     1.4235   2.222    0.029 *  
## zBetaAgency:zAgencyBonus        1.4501     2.9050   0.499    0.619    
## zBetaAgency:zAge               -1.4899     1.4688  -1.014    0.313    
## zAgencyBonus:zAge               1.8497     3.6647   0.505    0.615    
## zBetaAgency:zAgencyBonus:zAge   1.7630     2.6769   0.659    0.512    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.15 on 83 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.07712,    Adjusted R-squared:  -0.0007151 
## F-statistic: 0.9908 on 7 and 83 DF,  p-value: 0.4437

LOC

## 
## Call:
## lm(formula = LOC ~ zAge, data = LOC)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.4335 -3.3923 -0.4242  3.4805 10.1914 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  12.6288     0.4372  28.886   <2e-16 ***
## zAge          0.2453     0.4392   0.559    0.578    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.17 on 89 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.003494,   Adjusted R-squared:  -0.007703 
## F-statistic: 0.3121 on 1 and 89 DF,  p-value: 0.5778

## 
## Call:
## lm(formula = LOC ~ zBetaAgency * zAgencyBonus * zAge, data = LOC)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.7161 -2.9065 -0.3207  2.9217  9.7008 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                    12.6246     0.4987  25.315   <2e-16 ***
## zBetaAgency                    -0.9811     0.5369  -1.827   0.0712 .  
## zAgencyBonus                   -1.0330     1.3031  -0.793   0.4302    
## zAge                            0.6066     0.4829   1.256   0.2126    
## zBetaAgency:zAgencyBonus       -0.2208     1.0007  -0.221   0.8259    
## zBetaAgency:zAge                0.3645     0.5006   0.728   0.4686    
## zAgencyBonus:zAge               0.6135     1.2310   0.498   0.6195    
## zBetaAgency:zAgencyBonus:zAge   0.6690     0.9099   0.735   0.4643    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.15 on 83 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.07987,    Adjusted R-squared:  0.002267 
## F-statistic: 1.029 on 7 and 83 DF,  p-value: 0.4171

BDI

## 
## Call:
## lm(formula = zBDI ~ zAge, data = BDI)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.94728 -0.78671 -0.01517  0.72806  2.78555 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.122e-16  1.042e-01   0.000    1.000
## zAge        3.587e-02  1.048e-01   0.342    0.733
## 
## Residual standard error: 0.9993 on 90 degrees of freedom
## Multiple R-squared:  0.001301,   Adjusted R-squared:  -0.009796 
## F-statistic: 0.1172 on 1 and 90 DF,  p-value: 0.7329

## 
## Call:
## lm(formula = zBDI ~ zBetaAgency * zAgencyBonus * zAge, data = BDI)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8539 -0.6755 -0.0233  0.6917  2.5970 
## 
## Coefficients:
##                                Estimate Std. Error t value Pr(>|t|)
## (Intercept)                    0.002931   0.120219   0.024    0.981
## zBetaAgency                    0.013087   0.129916   0.101    0.920
## zAgencyBonus                   0.074662   0.314943   0.237    0.813
## zAge                          -0.005686   0.116933  -0.049    0.961
## zBetaAgency:zAgencyBonus      -0.005344   0.241141  -0.022    0.982
## zBetaAgency:zAge              -0.160374   0.121634  -1.318    0.191
## zAgencyBonus:zAge             -0.157728   0.299366  -0.527    0.600
## zBetaAgency:zAgencyBonus:zAge -0.166452   0.220803  -0.754    0.453
## 
## Residual standard error: 1.016 on 84 degrees of freedom
## Multiple R-squared:  0.03632,    Adjusted R-squared:  -0.04399 
## F-statistic: 0.4523 on 7 and 84 DF,  p-value: 0.866

STAI

## 
## Call:
## lm(formula = zSTAI_t ~ zAge, data = STAI)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.82245 -0.96538  0.01261  0.83118  2.16747 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.001085   0.104658   0.010    0.992
## zAge        0.060134   0.106243   0.566    0.573
## 
## Residual standard error: 0.9982 on 89 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.003587,   Adjusted R-squared:  -0.007609 
## F-statistic: 0.3204 on 1 and 89 DF,  p-value: 0.5728

## 
## Call:
## lm(formula = zSTAI_t ~ zBetaAgency * zAgencyBonus * zAge, data = STAI)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8159 -0.9588  0.0664  0.8286  1.7670 
## 
## Coefficients:
##                                Estimate Std. Error t value Pr(>|t|)
## (Intercept)                    0.010483   0.122103   0.086    0.932
## zBetaAgency                   -0.057149   0.131628  -0.434    0.665
## zAgencyBonus                  -0.068192   0.316929  -0.215    0.830
## zAge                           0.089023   0.120037   0.742    0.460
## zBetaAgency:zAgencyBonus       0.017827   0.242557   0.073    0.942
## zBetaAgency:zAge              -0.054557   0.124118  -0.440    0.661
## zAgencyBonus:zAge              0.085548   0.301177   0.284    0.777
## zBetaAgency:zAgencyBonus:zAge  0.009472   0.222193   0.043    0.966
## 
## Residual standard error: 1.022 on 83 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.02576,    Adjusted R-squared:  -0.05641 
## F-statistic: 0.3135 on 7 and 83 DF,  p-value: 0.946

## 
## Call:
## lm(formula = zSTAI_s ~ zAge, data = STAI)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9203 -0.6732 -0.1498  0.4769  3.1426 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.937e-16  1.033e-01   0.000    1.000
## zAge        1.368e-01  1.038e-01   1.318    0.191
## 
## Residual standard error: 0.9905 on 90 degrees of freedom
## Multiple R-squared:  0.01894,    Adjusted R-squared:  0.008035 
## F-statistic: 1.737 on 1 and 90 DF,  p-value: 0.1909

## 
## Call:
## lm(formula = zSTAI_s ~ zBetaAgency * zAgencyBonus * zAge, data = STAI)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.5559 -0.6835 -0.1344  0.6615  2.8213 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)  
## (Intercept)                    0.08413    0.11552   0.728   0.4685  
## zBetaAgency                   -0.04869    0.12483  -0.390   0.6975  
## zAgencyBonus                   0.26794    0.30262   0.885   0.3785  
## zAge                           0.20466    0.11236   1.822   0.0721 .
## zBetaAgency:zAgencyBonus       0.39754    0.23171   1.716   0.0899 .
## zBetaAgency:zAge              -0.09125    0.11688  -0.781   0.4372  
## zAgencyBonus:zAge             -0.17596    0.28766  -0.612   0.5424  
## zBetaAgency:zAgencyBonus:zAge -0.12508    0.21217  -0.590   0.5571  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9764 on 84 degrees of freedom
## Multiple R-squared:  0.1102, Adjusted R-squared:  0.03608 
## F-statistic: 1.487 on 7 and 84 DF,  p-value: 0.1831